The present invention relates to a process for manufacturing hollow one-piece metal elements having a highly curved lateral wall, in particular, copper or copper alloy elements for manufacturing the resonating cavities of nuclear accelerators. Here and hereinafter, the term "one-piece element" is intended to mean an element formed in one piece with no joints of any kind. Numerous technical applications, a highly complex one of which is the manufacture of resonating cavities for nuclear accelerators, are known to require hollow elements involving a high degree of precision and surface finish. Resonating cavities, for example, consist of a number of substantially ellipsoidal or paraboloidal cells terminated at opposite ends by cylindrical mouths or "irises" coaxial with the cell axis. At present, each cell is formed from two bowl-shaped half cells drawn from copper or copper alloy sheet and welded together along the maximum diameter line perpendicular to the cell axis through the irises. For ensuring a high degree of dimensional accuracy and optimum surface finish (no blow holes, cracks, inclusions, oxidation, etc.), the two half cells must be welded using fairly sophisticated equipment, e.g. electron-beam or similar, which nonetheless still involves a certain number of rejects. Known methods of manufacturing hollow elements to a high degree of precision and surface finish, and involving electron-beam welding or similar of drawn half cells, therefore involve high production costs; fail to safeguard against manufacturing defects; result in a highly complex production process; and require considerable space, mainly due to the welding equipment employed.